Spectra of the difference, sum and product of idempotents (Q2773387)

The main result of the paper under review is that, if \({\mathcal A}\) is a unital Banach algebra and \(p,q\in{\mathcal A}\) satisfy \(p^2=p\) and \(q^2=q\), then NEWLINE\[NEWLINE\begin{aligned} \sigma(pg)\setminus\{0,1\} &=\{1-\mu^2\mid\mu\in\sigma(p-q)\setminus\{-1,0,1\}\}\cr &=\{(1-\mu)^2\mid\mu\in\sigma(p+q)\setminus\{0,1,2\}\},\cr \end{aligned}NEWLINE\]NEWLINE where \(\sigma(\cdot)\) denotes the spectra of elements in \({\mathcal A}\). This extends a result of \textit{Matjaz Omladic} [Proc. Am. Math. Soc. 99, 317-318 (1987; Zbl 0636.47006)]. Some related results are also included, concerning elements of \({\mathcal A}\) which annihilate polynomials of degree 2.